Understanding Radial Velocity Variations in Barnard's Star Through Phase-Lag Analysis

M dwarfs are ideal targets to search for Earth-like planets in the habitable zones of nearby stars using the radial velocity method. Magnetic activity in M dwarfs (manifested as photospheric features such as star spots) remains a major challenge for exoplanet detection, since it can generate radial velocity (RV) signals that mimic planetary signals. In particular, the phase lag between stellar activity indicators and RV variations is a sign of spectral line asymmetries. Barnard's Star (GJ 699), a nearby and magnetically quiet M dwarf, has been extensively monitored by the ESPRESSO GTO program. By analyzing the phase lag between the stellar activity indicator FWHM and the RVs from the 157 ESPRESSO observations of Barnard's Star collected over four years, we aim to better understand the role of star spots in the observed radial velocity variations. We used Welch's method to measure the coherence between FWHM and RV, from which the phase spectrum was estimated. To explore the relationship between the phase lag amplitude and the correlation of both time series, we performed experiments using Gaussian Process (GP) models shifted by different time offsets to simulate a range of phase lags, allowing us to account for the uneven sampling and gaps present in the ESPRESSO observations. The phase spectrum shows a clear correlation with an increasing linear trend, suggesting a phase lag of about 48 days. Our experiments indicate that the correlation between FWHM and RV varies periodically, possibly due to star spots modulated by stellar rotation. Understanding this phase relation provides valuable information on how stellar surface inhomogeneities affect radial velocity measurements, improving our ability to distinguish stellar activity from true planetary signals.